On maximality of some solvable and locally nilpotent subalgebras of the Lie algebra $W_n(K)$

Let $K$ be an algebraically closed field of characteristic zero, $P_n=K[x_1,\mathrm{.}\mathrm{.}\mathrm{.},x_n]$ the polynomial ring, and $W_n(K)$ the Lie algebra of all $K$-derivations on $P_n$. One of the most important subalgebras of $W_n(K)$ is the triangular subalgebra $u_n(K)=P_0{\mathrm{\partial }}_1+\cdots +P_{n-1}{\mathrm{\partial }}_n$, where ${\mathrm{\partial }}_i:=\mathrm{\partial }\mathrm{/}\mathrm{\partial }{x}_i$ are partial derivatives on $P_n$. This subalgebra consists of locally nilpotent derivations on $P_n\mathrm{.}$ Such derivations define automorphisms of the ring $P_n$ and were studied by many authors. The subalgebra $u_n(K)$ is contained in another interesting subalgebra $s_n(K)=(P_0+x_1{P}_0){\mathrm{\partial }}_1+\cdots +(P_{n-1}+x_n{P}_{n-1}){\mathrm{\partial }}_n,$ which is solvable of the derived length $2n$ that is the maximum derived length of solvable subalgebras of $W_n(K)\mathrm{.}$ It is proved that $u_n(K)$ is a maximal locally nilpotent subalgebra and $s_n(K)$ is a maximal solvable subalgebra of the Lie algebra $W_n(K)\mathrm{.}$